Modular Forms and Sums of Squares
This post is an elaboration of an example (Example 1.2.1) in …
An interesting question is how many ways are there to represent a given number as a sum of squares. First we could consider representing a number as the sum of two squares. If we consider the number 5, we find that there are 8 possible ways when you consider order and sign.
5 = (±2)2 + (±1)2 = (±1)2 + (±2)2
More formally we can define Sn(k) to be the number of ways to represent the number k as a sum of n squares. From the example above, S2(5) = 8. Formulas for the small values of n have been found using classical methods of number theory. For instance
$$ S_2 (k) = 2 \left( 1 + \left(\frac{-1}{k}\right) \right) \sum_{d | k} \left(\frac{-1}{d}\right) $$
where $ () $ is the Jacobi symbol.
To understand the Jacobi symbol, we first need to understand the Legendre symbol, and to understand the Legendre symbol it is necessary to know a bit about quadratic residues.